Principle of Mathematical Induction

Q1.Prove: 11ⁿ⁺² + 12²ⁿ⁺¹ is divisible by 133 for all n ≥ 1. (Note: 133 = 7 × 19.) This is an example of:

• AA simple computation
• BA complex divisibility proof using PMI
• CA statement provable only by direct computation
• DA false statement

Q2.The sum 1 + 2 + 3 + ... + n equals:

• An(n+1)/4
• Bn(n–1)/2
• Cn(n+1)/2
• D(n+1)(n+2)/2

Q3.The method used to find the formula for a sum before proving it by PMI is called:

• ADeduction
• BConjecture (pattern recognition)
• CContradiction
• DContrapositive

Q4.Prove: For all n ≥ 1, (1+x)ⁿ ≥ 1 + nx for x > –1. This is known as:

• ACauchy inequality
• BBernoulli's inequality
• CAM-GM inequality
• DChebyshev's inequality

Q5.Prove: 1·2·3 + 2·3·4 + ... + n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4. In the inductive step, we add (k+1)(k+2)(k+3) to the hypothesis and get:

• Ak(k+1)(k+2)(k+3)/4 + (k+1)(k+2)(k+3) = (k+1)(k+2)(k+3)(k+4)/4
• Bk(k+1)(k+2)(k+3)/4 + k(k+1)(k+2) = (k+1)(k+2)(k+3)(k+4)/4
• Ck(k+1)(k+2)(k+3)/4 + (k+1)(k+2)(k+3) = k(k+1)(k+2)(k+4)/4
• D(k+1)(k+2)(k+3)(k+4) / 3